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Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

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Abstract

We consider a nonlinear Schrödinger equation in a time-dependent domain Q τ of ℝ2 given by

$$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$

We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.

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Acknowledgements

Mauricio Sepúlveda thanks the support of Fondecyt project 1110540, CONICYT project Anillo ACT1118 (ANANUM), and Basal, CMM, Universidad de Chile. Octavio Vera thanks the support of Fondecyt project 1121120. V. Bisognin, C. Buriol and Marcio Ferreira thank the support of Fundação de Amparo à pesquisa do Estado do Rio Grande do Sul. FAPERGS.

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Authors and Affiliations

  1. Centro Universitario Franciscano, 97010-032, Santa Maria, RS, Brazil

    Vanilde Bisognin

  2. Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil

    Celene Buriol & Marcio V. Ferreira

  3. CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile

    Mauricio Sepúlveda

  4. Departamento de Matemática, Universidad del Bío-Bío, Collao 1202, Casilla 5-C, Concepción, Chile

    Octavio Vera

Authors
  1. Vanilde Bisognin
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  2. Celene Buriol
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  3. Marcio V. Ferreira
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  4. Mauricio Sepúlveda
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  5. Octavio Vera
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Correspondence to Mauricio Sepúlveda.

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Bisognin, V., Buriol, C., Ferreira, M.V. et al. Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries. Acta Appl Math 125, 159–172 (2013). https://doi.org/10.1007/s10440-012-9785-0

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  • DOI: https://doi.org/10.1007/s10440-012-9785-0

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